Some special properties of dimension 8, in addition to the ones you identify:

• Bernstein's problem holds up to dimension $n=8$. The only function of $\mathbb{R}^{n-1}$ whose graph in $\mathbb{R}^n$ is minimal is a linear function. This fails in dimension $n=9$, with failure due to the existence of the Simons cone in dimension 8, so it's related to your last bullet point.

• There are 4 infinite families of Euclidean reflection groups, with exceptional ones only up to dimension 8. This is related to the existence of the exceptional simplex reflection groups and exceptional Lie algebras.

• There are 4 infinite families of holonomy groups of Riemannian manifolds, with two exceptional cases of $G_2$ and $Spin(7)$, the latter being in dimension 8.

• As pointed out by @YCor, triality holds for $Spin(8)$. $Spin(8)$ has three 8-dimensional irreducible representations which are permuted by the $S_3$ action associated with the symmetries of the $D_4$ Dynkin diagram.

• Cohn and Kumar found various tight simplices including a maximal 15 point tight simplex in $\mathbb{HP}^2$ which is 8 dimensional. A simplex in this case refers to a collection of equidistant points.

There are several other examples in the comments of phenomena where 8 dimensions is the first dimension in which the phenomenon appears (or is known to appear), but I've listed examples that seem to be special to dimension 8 (and most seem to be connected to the phenomena that you've already identified).